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<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> ;<br/>

<b>set </b><font color="Maroon" title="c7">nx</font> =  <a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c5">x</font>;<br/>
<b>set </b><font color="Maroon" title="c8">ny</font> =  <a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c6">y</font>;<br/>
<b>set </b><font color="Maroon" title="c9">dx</font> =  <a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font>;<br/>
<b>set </b><font color="Maroon" title="c10">dy</font> =  <a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font>;<br/>
<a NAME="E1:87_1_1"/>
(  <a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  &amp;  <a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T41" target="_self" title="ARYTM_3:th.41">Th41</a></i>;<br/>
<a NAME="E2:87_1_1"/><b>then </b><i><font color="Green" title="E74">A1</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">/</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">/</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T50" target="_self" title="ARYTM_3:th.50">Th50</a>, <a class="ref" href="ordinal3.html#T34" title="ORDINAL3:th.34">ORDINAL3:34</a></i>;<br/>
<a NAME="E3:87_1_1"/><i><font color="Green" title="E75">A2</font></i>: 
( <span class="p1">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">/</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; <span class="p1">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">/</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T45" target="_self" title="ARYTM_3:th.45">Th45</a></i>;<br/>
<a NAME="E4:87_1_1"/>
( <span class="p1">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font></span>)</span> or <span class="p1">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font></span>)</span> )
 
;<br/>
<a NAME="E5:87_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> or <span class="p1">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal3.html#D6" title="ORDINAL3:def.6">ORDINAL3:def 6</a></i>;<br/>
<a NAME="E6:87_1_1"/><b>then </b>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p4">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p4">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p4">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="arytm_3.html#K8" title="ARYTM_3:func.8">numerator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p4">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9" title="ARYTM_3:func.9">denominator</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E2:87_1_1"><i><font color="Green" title="E74">A1</font></i></a>, <a class="txt" href="arytm_3.html#E3:87_1_1"><i><font color="Green" title="E75">A2</font></i></a>, <a class="ref" href="arytm_3.html#T53" target="_self" title="ARYTM_3:th.53">Th53</a></i>;<br/>
<b>hence </b><a NAME="E7:87_1_1"/>
( ( for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>   holds not <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b1">z</font> ) implies  ex <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  st <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b1">z</font> )
 ;<br/>


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